Need for the advent of Calculus in India: A Journey into ... · Need for the advent of Calculus in India: A Journey into the 15th Century K. Ramasubramanian Cell for Indian Science

Need for the advent of Calculus in India:A Journey into the 15th Century

K. RamasubramanianCell for Indian Science and Technology in Sanskrit

Department of HSS, IIT Bombay

August 30, 2008BMS Engineering College, Bangalore

Organized by: Sri Tirunarayana Trust

1

IntroductionWhat is Calculus ?

◮ Calculus is basically study of how things change.

◮ Fundamental idea : Study the change by studying the“instantaneous” change. (Position of Sun/Moon after one hour)

◮ A typical course in calculus includes:

◮ How to find the instantaneous change ? (Differentiation !)◮ How to use derivatives to solve problems ?◮ How to get back the function from the derivative of the

function ? (Integration !)◮ Expressing functions in terms of Infinite series◮ Study of the convergence of series, and so on....

◮ These involve the notion of “infinitesmal” and “infinity”.

Unlike Greece, India never had a fear of the infinite orof the void. Indeed it embraced them.1

1Charles Seife, Zero:The Biography of a Dangerous Idea, p.20.2

Outline

◮ Introduction◮ Zero and Infinity◮ N ılakan. t.ha’s discussion of irrationality of π

◮ Sum of an infinite geometric series◮ San.kara’s discussion of the binomial series expansion◮ Estimation of sums of powers of integers 1 to n for large n◮ Derivation of the Madhava series for π

◮ Derivation of end-correction terms (Antya-sam. skara)◮ Madhava’s series for Rsine and Rversine◮ Instantaneous velocity and derivatives◮ Concluding Remarks

3

IntroductionBroad classification of Knowledge – Mun. d. aka-upanis.ad

Why do I thinkthe way I think ?

Body of knowledge

Veda

Para Apara

Samaveda

Atharvaveda

Rgveda

Yajurveda

Siksha

Kalpa

Vyakarana

Nirukta

Chanda

Jyotisham

Who am I ?

Why did I come into existence ?

Fudamental

and eternal

questions ?

4

IntroductionCelestial Sphere

◮ Great thinkers of all the civilizations – Hindu, Greek,Arabic2, Chinese, etc. – wondered how to interpret thecelestial phenomena.

2Nasir al-Din al-Tusi, Ibn al-Shatir, . . .5

IntroductionZero and Infinity: ZUa:nya and A:na:nta

ESSENCE OF CALCULUS ≡ Use of infinitesmals/limits3

Greeks could not do this neat little mathematical trick. They didn’thave the concept of a limit because they didn’t believe in zero. Theterms in the infinite series didn’t have a limit or a destination; theyseemed to get smaller and smaller without any particular end in sight.As a result the Greeks couldn’t handle the infinite. They pondered theconcept of void but rejected zero as a number, and they toyed withthe concept of infinite but refused to allow infinity – numbers that areinifinitely small and infinitely large – anywhere near the realm ofnumbers. This is the biggest failure in the Greek Mathematics, and itis the only thing that kept them from discovering calculus. 4

3One of the passages to “limit” is by summing an infinite series.4Charles Seife, Zero:The Biography of a Dangerous Idea, Viking, 2000;

Rupa & Co. 2008.6

Fear of zero!

◮ To the ancients,5 zero’s mathematical properties wereinexplicable . . . because zero is different from other numbers. . . zero always misbehaves. At the very least it does not behavethe way the other numbers do.

◮ Add any number to itself, and it changes. (1 + 1 = 2)◮ Zero refuses to get bigger. (0 + 0 = 0 ?)6

◮ It also refuses to make any other number bigger.7

◮ Normally multiplication by a number stretches a number line. Butmultiplication by zero collapes it.

◮ Dividing by zero destroys the entire framework of mathematics.. . . it would clash with the fundamental philosophy of the west.8

5particularly Greeks.6This violates the basic principle of numbers called the Axiom of

Archimedes, which says that if you add something to itself enought times, itwill exceed any other number in magnitude.

7Charles Seife, Zero:The Biography of a Dangerous Idea, p.20.8Ibid. p.23.

7

IntroductionThe Infinitesmal and Infinity

◮ This being the scenario in the Greek tradition, it is interesting tocontrast it with the Indian tradition.

◮ the dexterity with which the Hindus could comprehend suchconcepts is evident from the santi-mantra of the Isavasyopanis.adthat runs as follows::pUa:NRa:ma:dH :pUa:NRa:�a.ma:dM :pUa:Na.Ra:t,a :pUa:NRa:mua:d:. ya:tea Á:pUa:NRa:~ya :pUa:NRa:ma.a:d.a:ya :pUa:NRa:mea:va.a:va:�a.Za:Sya:tea Á Á

. . . when purn. a is taken out of purn. a what remains isalso purn. a.

◮ The notion of sunya appears Chandassutra (c.300 BC).9

◮ Brahmagupta taking about the mathematics of zero observes:;Da:nRa:NRa:ya.ea:DRa:nMa �+Na:mxa:Na:ya.eaH ;Da:na:NRa:ya.ea.=;nta.=M .sa:mEa:k�+.a:m,a Á�+Na:mEa:k�+.aM .. a ;Da:na:mxa:Na:Da:na:ZUa:nya:ya.eaH ZUa:nya:m,a Á Á 10

9 .�+pMa ZUa:nyea Á ;�a.dõ H ZUa:nyea (8.29,30).10Brahmasphut.a-siddhanta – 18.30.8

IntroductionThe Infinitesmal and Infinity

The notion of infinity, which is so fundamental for the development ofcalculus besides with the notion of infinitesimal, as presented byBhaskara in his Bıjagan. ita:A:�///////�a.sma:n,a ;�a.va:k+a.=H Ka:h:=e na .=:a:Za.Ea A:�a.pa :pra:�a.va:�e :Sva:�a.pa ;a.naHsxa:tea:Sua Ába:hu :Sva:�a.pa .~ya.a:t,a l+ya:sxa:�a.�:k+a:le A:na:nteaY:. yua:tea BUa:ta:ga:Nea:Sua ya:dõ :t,a Á Á

In this quantity that has zero as the divisor (khahara) thereis no change, even if very large quantities [numbers of hugemagnitudes] are inserted or extracted; as no change takesplace in the infinite (ananta) immutable (acyuta) [Brahman]at the time of destruction or creation of the worlds whennumerous beings are absorbed and put forth.

9

IntroductionSignal achievements of Kerala Mathematicians

◮ Infinite series expansions for trigonometric functions

sin x = x −x3

3!+

x5

5!− . . . , (1)

◮ Infinite series for π

Paridhi = 4 × Vyasa ×

(

1 −13

+15−

17

+ . . .

)

(2)

◮ The derivative of sine inverse function

ddt

[

sin−1( r

Rsin M

)]

=

rR

cos MdMdt

√

1 −( r

R sin M)2

(3)

10

IntroductionNeed for the precise values of Sines and Derivatives

◮ In Indian astronomical texts the sine function is describedby the term jya or jıva

◮ This function is almost ubiquitous. For instance,◮ In the computation of longitude of the planets,

θ = θ0 − sin−1( r

Rsin M

)

(4)

◮ The declincation of the Sun is computed using the formula,

sin δ = sin ǫ sin λ, (5)

where ǫ → obliquity of the ecliptic and δ → declination ofthe Sun.

◮ Given that the sine function appears in almost all thecomputations, a procedure to compute its precise valuebecomes all the more important.

11

N ılakan. t.ha’s discussion of irrationality of π

◮ While discussing the value of π Nılakan. t.ha observes::pa:�a=;a.Da:v.ya.a:sa:ya.eaH .sa:*ñÍËÉ ùÁ+;a.a-.sa:}ba:nDaH :pra:d:�a.ZRa:taH Á . . .A.a:sa:�aH, A.a:sa:�a:ta:yEa:va A:yua:ta:dõ :ya:sa:*ñÍËÉ ùÁ+;a:�a.va:Sk+.}Ba:~ya I+yMa:pa:�a=;a.Da:sa:*ñÍËÉ ùÁ+;a.a o+�+a Á ku +.taH :pua:naH va.a:~ta:v�a.Ma .sa:*ñÍËÉ ùÁ+;a.a:m,a o+tsxa.$yaA.a:sa:�Ea:va I+h.ea:�+a ? o+. ya:tea Á ta:~ya.a va:�u +.ma:Za:k�+.a:tva.a:t,a Á ku +.taH ?

The relation between the circumference and the diameterwas expressed. . . .Approximate: This value (62,832) was stated to be nearlythe circumference of a circle having a diameter of 20,000.“Why then has an approximate value been mentioned hereleaving behind the actual value?” It is explained [asfollows]. Because it (the exact value) cannot be expressed.Why?

12

N ılakan. t.ha’s discussion of irrationality of πyea:na ma.a:nea:na m�a.a:ya:ma.a:na.ea v.ya.a:saH ;a.na.=;va:ya:vaH .~ya.a:t,a, .tea:nEa:va m�a.a:ya:ma.a:naH:pa:�a=;a.DaH :pua:naH .sa.a:va:ya:va O;:va .~ya.a:t,a Á yea:na .. a m�a.a:ya:ma.a:naH :pa:�a=;a.DaH;a.na.=;va:ya:vaH .tea:nEa:va m�a.a:ya:ma.a:na.ea v.ya.a:sa.eaY:�a.pa .sa.a:va:ya:va O;:va; I+a.ta O;:ke +.nEa:vam�a.a:ya:ma.a:na:ya.eaH o+Ba:ya.eaH ë�ÅëÁ*:+a:�a.pa na ;a.na.=;va:ya:va:tvMa .~ya.a:t,a ÁGiven a certain unit of measurement (mana) in terms of whichthe diameter (vyasa) specified [is just an integer and] has no[fractional] part (niravayava), the same measure whenemployed to specify the circumference (paridhi) will certainlyhave a [fractional] part (savayava) [and cannot be just aninteger]. Again if in terms of certain [other] measure thecircumference has no [fractional] part, then employing the samemeasure the diameter will certainly have a [fractional] part [andcannot be an integer]. Thus when both [the diameter and thecircumference] are measured by the same unit, they cannotboth be specified [as integers] without [fractional] parts.

13

N ılakan. t.ha’s discussion of irrationality of πma:h.a:nta:m,a A:Dva.a:nMa ga:tva.a:�a.pa A:�pa.a:va:ya:va:tva:m,a O;:va l+Bya:m,a Á;a.na.=;va:ya:va:tvMa tua ë�ÅëÁ*:+a:�a.pa na l+Bya:m,a I+a.ta Ba.a:vaH ÁEven if you go a long way (i.e., keep on reducing themeasure of the unit employed), the fractional part [inspecifying one of them] will only become very small. Asituation in which there will be no [fractional] part (i.e,both the diameter and circumference can be specifiedin terms of integers) is impossible, and this is what isthe import [of the expression asanna]

What Nılakan. t.ha is trying to explain is the incommensurabilityof the circumference and the diameter of a circle. The last lineof the above quote – however small you may choose your unitof measurement to be, the two quantities will never becomecommensurate – is indeed noteworthy.

14

Sum of an infinite geometric series

◮ In his Aryabhat. ıya-bhas.ya, while deriving an interestingapproximation for the arc of circle in terms of the jya(Rsine) and the sara (Rversine), Nılakan. t.ha presents adetailed demonstration of how to sum an infinite geometricseries.

◮ The specific geometric series that arises in the abovecontext is:

14

+

(

14

)2

+ . . . +

(

14

)n

+ . . . =13

.

◮ Here, we shall present an outline of Nılakan. t.ha’s argument◮ It is clearly indicative of how the notion of limit was

understood.

15


◮ AB is capa (c) as itlooks like a bow.

◮ AD is jyardha (j) as ithalf the string.

◮ BD is sara (s) as itlooks like an arrow.

O

A C

B

D

E

F

The expression given by Nılakan. t.ha is:

c ≈

√

(

1 +13

)

s2 + j2. (6).sa.�yMa:Za.a:�a.d:Sua:va:ga.Ra:t,a .$ya.a:va:ga.Ra:Q.�a.a:t,a :pa:dM ;Da:nuaH :pra.a:yaH Á16


The proof of (6) presented by Nılakan. t.ha involves:

1. Repeated halving of the arc-bit, capa c to get c1 . . . ci .

2. Finding the corresponding semi-chords, jya (ji ) and theRversines, sara (si )

3. Estimating the difference between the capa and jya at eachstep.

If ∆i be the difference between the capa and jya at the i th step,

∆i = ci − ji . (7)

Nılakan. t.ha observation – as the size of the capa decreases thedifference also decreases.

17

Sum of an infinite geometric seriesta.�a .$ya.a:. a.a:pa:ya.ea.=;nta.=;~ya :pua:naH :pua:naH nyUa:na:tvMa.. a.a:pa:pa:�a=;ma.a:Na.a:�pa:tva:kÒ +.mea:Nea:a.ta ta.�a-d:DRa:. a.a:pa.a:na.a:m,a A:DRa.$ya.a:pa.=;}å.pa.=:aZa.=;pa.=;}å.pa.=:a .. a A.a:n�a.a:ya:ma.a:na.a na ë�ÅëÁ*:+.a.. a:d:�a.pa :pa:yRa:va:~ya:a.ta A.a:na:ntya.a:d,;�a.va:Ba.a:ga:~ya Á ta:taH ;�a.k+.ya:nta:aúãÁ*.a:t,a :pra:de :ZMa ga:tva.a .. a.a:pa:~ya .j�a.a:va.a:ya.a:(ãÉaA:�p�a.a:ya:~tva:m,a A.a:pa.a:dùÅ;a .. a.a:pa.$ya.a:nta.=M .. a ZUa:nya:pra.a:yMa l+b.Dva.a :pua:na.=;�a.pak+.�pya:ma.a:na:ma:nta.=;m,a A:tya:�pa:ma:�a.pa k+Ea:Za:l;a:t,a ¼ea:ya:m,a Á◮ Generating successive values of the jis and sis is an

“unending” process as one can keep on dividing the capainto half ad infinitum.

◮ It would therefore be appropriate to recognize that thedifference ∆i is tending to zero and hence make an“intelligent approximation”, to obtain the value of thedifference between c and j approximately.

18

Sum of an infinite geometric seriesNılakan. t.ha poses a very important question:k+.TMa :pua:naH ta.a:va:de :va va:DRa:tea ta.a:va:dõ :DRa:tea .. a ?

How do you know that [the sum of the series]increases only upto that [limiting value] and thatcertainly increases upto that [limiting value]?

Proceeding to answer he first states the general result

a

[

(

1r

)

+

(

1r

)2

+

(

1r

)3

+ . . .

]

=a

r − 1. (8)

◮ Infinite Geometric Series – tua:�ya:. Ce +d:pa.=;Ba.a:ga:pa.=;}å.pa.=:a◮ Common Divisor – Ce +d

19

Sum of an infinite geometric seriesHe further notes that this result is best demonstrated with r = 4and obtains the sequence of results,

13

=14

+1

(4.3),

1(4.3)

=1

(4.4)+

1(4.4.3)

,

1(4.4.3)

=1

(4.4.4)+

1(4.4.4.3)

, (9)

and so on, which leads to the general result,

13−

[

14

+

(

14

)2

+ . . . +

(

14

)n]

=

(

14

)n (13

)

. (10)

As we sum more terms, the difference between 13 and sum of

powers of 14 , becomes extremely small, but never zero.

20

What is a Limit ?

Cauchy’s (1821) definition of limit:

If the successive values attributed to the samevariable approach indefinitely a fixed value, such thatfinally they differ from it by as little as one wishes, thislatter is called the limit of all the others.11

Nılakan. t.ha in his Aryabhat. ıya-bhas.ya:k+.TMa :pua:naH ta.a:va:de :va va:DRa:tea ta.a:va:dõ :DRa:tea .. a ?How do you know that [the sum of the series]increases only upto that [limiting value] and thatcertainly increases upto that [limiting value]?

11Cauchy, Cours d’Analyse, cited by Victor J. Katz, A History ofMathematics, Addison Wesley Longman, New York 1998, p. 708.

21

Binomial series expansion

Sankara Variyar in his Kriyakramakarı discusses as follows

◮ Consider the product a( c

b

)

◮ Here, a is called gun. ya, c the gun. aka and b the hara (theseare all assumed to be positive).

◮ If we consider the ratio( c

b

)

, there are two possibilities:◮ Case I: gun. aka > hara (c > b). In this case we rewrite the

product in the following form

a(c

b

)

= a + a(c − b)

b. (11)

◮ Case II: gun. aka < hara (c < b). In this case we rewrite theproduct as

a(c

b

)

= a − a(b − c)

b. (12)

22

Binomial series expansion

In the expression a (b−c)b , if we want to replace the division by b by

division by c, then we have to make a subtractive correction(sodhya-phala) which amounts to the following equation.

a(b − c)

b= a

(b − c)

c− a

(b − c)

c×

(b − c)

b. (13)

If we again replace the division by the divisor b by the multiplier c,

acb

= a −

[

a(b − c)

c− a

(b − c)

c×

(b − c)

b

]

= a −

[

a(b − c)

c− a

(b − c)

c×

(b − c)

c×

cb

]

= a −

[

a(b − c)

c−

[

a(b − c)2

c2 −

(

a(b − c)2

c2 ×(b − c)

b

)]]

(14)

The quantity a (b−c)2

c2 is called dvitıya-phala or simply dvitıya and theone subtracted from that is dvitıya-sodhya-phala.

23

Binomial series expansionThus, after taking m sodhya-phala-s we get

acb

= a − a(b − c)

c+ a

[

(b − c)

c

]2

− . . . + (−1)m−1a[

(b − c)

c

]m−1

+(−1)ma[

(b − c)

c

]m−1(b − c)

b. (15)O;:vMa mua:hu H :P+l;a:na:ya:nea kx +.teaY:�a.pa yua:a.�+.taH ë�ÅëÁ*:+a:�a.pa na .sa:ma.a:a.�aH Á ta:Ta.a:�a.pa ya.a:va:d:pea:[Ma.sUa:[ma:ta.a:ma.a:pa.a:dùÅ;a :pa.a:(ãÉa.a:tya.a:nyua:pea:[ya :P+l;a:na:ya:nMa .sa:ma.a:pa:n�a.a:ya:m,a ÁI+h.ea.�a.=:ea.�a.=;P+l;a:na.Ma nyUa:na:tvMa tua gua:Na:h.a.=:a:nta:=e gua:Na:k+a.=:a:�yUa:na O;:va .~ya.a:t,a Á

◮ Still, if we keep including correction terms, then there is logicallyno end to the series of correction terms (phala-parampara).

◮ For achieving a given level of accuracy, we can terminate theprocess when the correction term becomes small enough.

◮ If b − c < c, then the successive correction terms keepdecreasing.

24

Different approximations to π

◮ The Sulba-sutra-s, give the value of π close to 3.088.

◮ Aryabhat.a (499 AD) gives an approximation which is correct tofour decimal places... a:tua.=;a.Da:kM Za:ta:ma:�:gua:NMa dõ .a:Sa:�a.�:~ta:Ta.a .sa:h:~åò:a.a:Na.a:m,a ÁA:yua:ta:dõ :ya:�a.va:Sk+.}Ba:~ya.a:sa:�a.ea vxa.�a:pa:�a=;Na.a:hH Á Á

π =(100 + 4) × 8 + 62000

20000=

6283220000

= 3.1416

◮ Then we have the verse of Lılavatı12v.ya.a:sea Ba:na:nd.a:a.çÉîå+;a:h:tea ;�a.va:Ba:�e Ka:ba.a:Na:sUa:yERaH :pa:�a=;a.DaH .sua:sUa:[maH Ádõ .a:�a.vMa:Za:a.taÈåî ÁÁ*+e ;�a.va:&+teaY:Ta ZEa:lE H .~TUa:l;eaY:Ta:va.a .~ya.a:d, v.ya:va:h.a.=;ya.ea:gyaH Á Áπ =

39271250

= 3.1416 that’s same as Aryabhat.a’s value.

12Lılavatı of Bhaskaracarya, verse 199.25

Different approximations to π

The commentary Kriyakramakarı further proceeds to present moreaccurate values of π given by different Acaryas.ma.a:Da:va.a:. a.a:yRaH :pua:naH A:ta.ea:pya.a:sa:�a:ta:ma.Ma :pa:�a=;a.Da:sa:*ñÍö÷ÅÉ ÙùÅ+;a.a:mua:�+.va.a:n,a –;�a.va:bua:Da:nea.�a:ga.ja.a:�a.h:hu :ta.a:Za:na:�a�a:gua:Na:vea:d:Ba:va.a.=;Na:ba.a:h:vaH Ána:va:a.na:Ka:vRa:�a.ma:tea vxa:a.ta:�a.va:~ta:=e :pa:�a=;a.Da:ma.a:na:�a.ma:dM .ja:ga:du :bRua:Da.aH Á Á13

The values of π given by the above verses are:

π =355113

= 3.141592920353 (correct to 6 places)

π =2827433388233

9 × 1011 = 3.141592653592 (correct to 11 places)

The latter one is due to Madhava.13Vibudha=33, Netra=2, Gaja=8, Ahi=8, Hutasana=3, Trigun. a=3,

Veda=4, Bha=27, Varan. a=8, Bahu=2, Nava-nikharva=9 × 1011. (Theword nikharva represents 1011).

26

Infinite series for the π – as given in Yukti-dıpikav.ya.a:sea va.a:�a=;a.Da:a.na:h:tea .�+pa:&+tea v.ya.a:sa:sa.a:ga.=:a:�a.Ba:h:tea Á;�a�a:Za.=:a:�a.d ;�a.va:Sa:ma:sa:*ñÍËÉ ùÁ+;a.a:Ba:�+.m,a �+NMa .~vMa :pxa:Ta:k, kÒ +.ma.a:t,a ku +.ya.Ra:t,a Á ÁThe diameter multiplied by four and divided by unity (is found andstored). Again the products of the diameter and four are divided bythe odd numbers like three, five, etc., and the results are subtractedand added in order (to the earlier stored result).

◮ vyase varidhinihate → 4 × Diameter (varidhi)

◮ vis.amasankhyabhaktam → Divided by odd numbers

◮ trisaradi → 3, 5, etc. (bhutasankhya system)

◮ r. n. am. svam. → to be subtracted and added [successively]


(

1 −13

+15−

17

+ . . . . . .

)

27

Infinite series for the π

The triangles OPi−1Ci andOAi−1Bi are similar. Hence,

Ai−1Bi

OAi−1=

Pi−1Ci

OPi−1(16)

Similarly triangles Pi−1CiPi

and P0OPi are similar.Hence,

Pi−1Ci

Pi−1Pi=

OP0

OPi(17)

Pi

O S

P0

Ai−1

i−1P

B

A

i

np

kk

i−1

i

O S

E

N

Blown up version of this quadrant

i

i

C

28


From these two relations we have,

Ai−1Bi =OAi−1.OP0.Pi−1Pi

OPi−1.OPi

= Pi−1Pi ×OAi−1

OPi−1×

OP0

OPi

=( r

n

)

×r

ki+1×

rki

=( r

n

)

(

r2

kiki+1

)

. (18)

It is( r

n

)

that is refered to as khan. d. a in the text. The text alsonotes that, when the khan. d. a-s become small (or equivalently nbecomes large), the Rsines can be taken as the arc-bits itself.:pa:�a=;a.Da:Ka:Nq+~ya.a:DRa.$ya.a → :pa:�a=;DyMa:Za

i.e., Ai−1Bi → Ai−1Ai .29


Though the value of 18 th of the circumference has been obtained as

C8

=( r

n

)

[(

r2

k0k1

)

+

(

r2

k1k2

)

+

(

r2

k2k3

)

+ · · · +

(

r2

kn−1kn

)]

, (19)

there may not be much difference in approximating it by either of thefollowing expressions:

C8

=( r

n

)

[

(

r2

k20

)

+

(

r2

k21

)

+

(

r2

k22

)

+ · · · +

(

r2

k2n−1

)]

(20)

orC8

=( r

n

)

[(

r2

k21

)

+

(

r2

k22

)

+

(

r2

k23

)

+ · · · +

(

r2

k2n

)]

(21)

The difference between (21) and (20) will be( r

n

)

[(

r2

k20

)

−

(

r2

k2n

)]

=( r

n

)

[

1 −

(

12

)]

( k20 , k2

n = r2, 2r2)

=( r

n

)

(

12

)

(22)Ka:Nq+~ya A:�pa:tva:va:Za.a:t,a ta:d:nta.=M ZUa:nya:pra.a:ya:mea:va Á30


Thus we have,

C8

=

n∑

i=1

rn

(

r2

k2i

)

=

n∑

i=1

[

rn−

rn

(

k2i − r2

r2

)

+rn

(

k2i − r2

r2

)2

− . . .

]

=( r

n

)

[1 + 1 + . . . + 1]

−( r

n

)

(

1r2

)

[

( rn

)2+

(

2rn

)2

+ . . . +(nr

n

)2]

+( r

n

)

(

1r4

)

[

( rn

)4+

(

2rn

)4

+ . . . +(nr

n

)4]

−( r

n

)

(

1r6

)

[

( rn

)6+

(

2rn

)6

+ . . . +(nr

n

)6]

+ . . . . (23)

31


If we take out the powers of bhuja-khan. d. arn , the summations involved

are that of even powers of the natural numbers, namelyedadyekottara-varga-sankalita, 12 + 22 + ... + n2,edadyekottara-varga-varga-sankalita, 14 + 24 + ... + n4, and so on.Kerala astronomers knew that

n∑

i=1

ik ≈nk+1

k + 1, (24)

we arrive at the result

C8

= r(

1 −13

+15−

17

+ · · ·

)

, (25)

The above equation is given in the form


(

1 −13

+15−

17

+ · · · · · ·

)

32

Summation of series (sankalita) [Integral ?]Background

The Aryabhat. ıya of Aryabhat.a has the formula for the sankalita-s

S(1)n = 1 + 2 + · · · + n =

n(n + 1)

2

S(2)n = 12 + 22 + · · · + n2 =

n(n + 1)(2n + 1)

6

S(3)n = 13 + 23 + · · · + n3 =

[

n(n + 1)

2

]2

(26)

From these, it is easy to estimate these sums when n is large.Yuktibhas. a gives a general method of estimating thesama-ghata-sankalita

S(k)n = 1k + 2k + · · · + nk , (27)

when n is large. What is presents is a general method of estimation,which does make use of the actual value of the sum. So, theargument is repeated even for k = 1, 2, 3, although the result ofsummation is well known in these cases.

33

Summation of series (sankalita) [Integral ?]Kevala-sankalita or Mula-sankalita

The following is a citation from the translation of Yuktibhas. a (6.4.1):

“Now is described the methods of making the summations. At first,the simple arithmetical progression (kevala-sankalita) is described.This is followed by the summation of the products of equal numbers(squares). . . .Here, in this mula-sankalita (basic arithmetical progression), the finalbhuja is equal to the radius. The term before that will be one segment(khan. d. a) less. The next one will be two segments less. Here, if all theterms (bhuja-s) had been equal to the radius, . . . .Now, the smaller the segments, the more accurate (suks.ma) will bethe result. Hence, do the summation also by taking each segment assmall as an atom (an. u). Here, if it (namely, the bhuja or the radius) isdivided into parardha (a very large number) parts, to the bhuja

obtained by multiplying by parardha add one part in parardha andmultiply by the radius and divide by 2, and then divide by parardha.For, the result will practically be the square of the radius divided bytwo. . . . ”

34


The first summation, the bhuja-sankalita, may be written in the orderfrom the final bhuja to the first bhuja as

S(1)n =

(nrn

)

+

(

(n − 1)rn

)

+ .... +( r

n

)

. (28)

Now, conceive of the bhuja-khan. d. arn as being infinitesimal (an. u) and

at the same time as of unit-measure (rupa), so that the radius will bethe measure of n, the pada, or the number of terms. Then

S(1)n = n + (n − 1) + .... + 1. (29)

If each of the terms were of the measure of radius (n) then the sumwould be nothing but n2, the square of the radius. But only the firstterm is of the measure of radius, the next is deficient by one segment(khan. d. a), the next by two segments and so on till the last term whichis deficient by an amount equal to radius-minus-one segment.

S(1)n = n + [n − 1] + [n − 2].... + [n − (n − 2)] + [n − (n − 1)]

= n.n − [1 + 2 + ... + (n − 1)]. (30)

35


S(1)n = n + [n − 1] + [n − 2].... + [n − (n − 2)] + [n − (n − 1)]

= n.n − [1 + 2 + ... + (n − 1)].

When n is very large, the quantity to be subtracted from n2 ispractically (prayen. a) the same as S(1)

n , thus leading to the estimate

S(1)n ≈ n2 − S(1)

n , (31)

or, equivalently

S(1)n ≈

n2

2. (32)

It is stated that the result is more accurate, the smaller the size of thesegments (or equivalently the larger the value of n). Sankara Variyar

notes in his Kriyakramakarı:Ka:Nq+~ya.a:�pa:tvea .sa:tyea:va l+b.Da:~ya .sUa:[ma:ta.a .. a .~ya.a:t,a Á36

Summation of series (sankalita)Varga-sankalita

With the same convention that rn is the measure of the unit, the

bhuja-varga-sankalita (the sum of the squares of the bhuja-s) will be

S(2)n = n2 + (n − 1)2 + .... + 12. (33)

In above expression, each bhuja is multiplied by itself. If instead, weconsider that each bhuja is multiplied by the radius (n in our units)then that would give raise to the sum

n [n + (n − 1) + ... + 1] = n S(1)n . (34)

This sum is exceeds the bhuja-varga-sankalita by the amount

nS(1)n − S(2)

n = 1.(n − 1) + 2.(n − 2) + 3.(n − 3) + . . . + (n − 1).1.

This may be written as

nS(1)n − S(2)

n = (n − 1) + (n − 2) + (n − 3) + . . . +1

+(n − 2) + (n − 3) + . . . +1

+(n − 3) + . . . +1

+ . . . . (35)

37

Summation of series (sankalita)Varga-sankalita and Sankalita-sankalita

Thus,nS(1)

n − S(2)n = S(1)

n−1 + S(1)n−2 + S(1)

n−3 + . . . . (36)

For large n, we have already estimated S(1)n ≈ n2

2 . Thus,

nS(1)n − S(2)

n =(n − 1)2

2+

(n − 2)2

2+

(n − 3)2

2+ . . . . (37)

Thus, the right hand side of (36) (the sankalita-sankalita or the excess

of nS(1)n over S(2)

n ) is essentially S(2)n2 for large n, so that we obtain

nS(1)n − S(2)

n ≈S(2)

n

2. (38)

Again, using the earlier estimate (19) for S(1)n , we obtain the result

S(2)n ≈

n3

3. (39)

Thus bhuja-varga-sankalita is one-third the cube of the radius.38

Summation of series (sankalita)Samaghata-sankalita

Thus in general we have,

nS(k−1)n − S(k)

n ≈(n − 1)k

k+

(n − 2)k

k+

(n − 3)k

k+ . . .

≈

(

1k

)

S(k)n . (40)

Rewriting the above equation we have

S(k)n ≈ nS(k−1)

n −

(

1k

)

S(k)n . (41)

(A:ta o;�a.=:ea.�a.=;sa:ñÍö�ÅÅ*:+a.l+ta.a:na:ya:na.a:ya ta.�a:tsa:ñÍö�ÅÅ*:+a.l+ta:~ya v.ya.a:sa.a:DRa:gua:Na:na:m,aO;;kE +.k+a:a.Da:k+.sa:*ñÍËÉ ùÁ+;a.a:�a-.~va.Ma:Za:Za.ea:Da:nMa .. a k+a:yRa:m,a I+a.ta ;�////�a.~Ta:ta:m,a Á )Thus we obtain the estimate

S(k)n ≈

nk+1

(k + 1). (42)

39

End-correction in the infinite series for π

Expression for the “remainder” terms (Antyasam. skara)ya:tsa:*ñÍËÉ ùÁ+;a:ya.a.�a h.=;Nea kx +.tea ;a.na:vxa.�a.a &+a.ta:~tua .ja.a:�a.ma:ta:ya.a Áta:~ya.a �+.DvRa:ga:ta.a ya.a .sa:ma:sa:*ñÍËÉ ùÁ+;a.a ta:�;lM gua:Na.eaY:ntea .~ya.a:t,a Á Áta:dõ :ga.eRa .�+pa:yua:ta.ea h.a.=:ea v.ya.a:sa.a:�/�a.b.Da:Ga.a:ta:taH :pra.a:gva:t,a Áta.a:Bya.a:ma.a:�Ma .~va:mxa:Nea kx +.tea ;Ga:nea [ea:pa O;:va k+=;N�a.a:yaH Á Ál+b.DaH :pa:�a=;a.DaH .sUa:[maH ba:hu :kx +.tva.ea h.=;Na:ta.eaY:a.ta:sUa:[maH .~ya.a:t,a Á Á◮ yatsankhyayatra haran. e → Dividing by a certain number (p)

◮ nivr. tta hr. tistu → if the division is stopped

◮ jamitaya → being bored (due to slow-convergence)

Remainder term =

(

p+12

)

(

p+12

)2+ 1

◮ labdhah. paridhih. suks.mah. → the circumference obtained wouldbe quite accurate

40


When does the end-correction give exact result ?The discussion by Sankara Variyar is almost in the form of aengaging dialogue between the teacher and the taught andcommences with the question, how do you ensure accuracy.k+.TMa :pua:na.=:�a mua:hu :�a.vRa:Sa:ma:sa:*ñÍËÉ ùÁ+;a.a:h.=;Nea:na l+Bya:~ya :pa:�a=;DeaHA.a:sa:�a:tva:ma:ntya:sMa:~k+a:=e ;Na A.a:pa.a:dùÅ;a:tea? o+. ya:tea Áta.�a ta.a:va:du :�+�+pa:ssMa:~k+a.=H .sUa:[ma.ea na :vea:a.ta :pra:Ta:mMa ;a.na.�+pa:N�a.a:ya:m,a Áta:d:T a ya:ya.a:k+.ya.a-;a.. a:d, ;�a.va:Sa:ma:sMa:K.ya:ya.a h.=;Nea kx +.tea :pxa:Ta:k, .sMa:~k+a.=Mku +.ya.Ra:t,a Á A:Ta ta:du .�a.=;�a.va:Sa:ma:sMa:K.ya.a-h.=;Na.a:na:nta.=M .. a :pxa:Ta:k, .sMa:~k+a.=Mku +.ya.Ra:t,a Á O;:vMa kx +.tea l+b.Da.Ea :pa:�a=;D�a.a ya:�a.d tua:�ya.Ea Ba:va:taH ta:�a.hR .sMa:~k+a.=H.sUa:[ma I+a.ta ;a.na:N�a.Ra:ya:ta.a:m,a Á k+.Ta:m,a?

How do you say that you get the value close to thecircumference by using antya-sam. skara, instead ofrepeatedly dividing by odd numbers? Let me explain this.

41


When does the end-correction give exact result ?

The argument is as follows: If the correction term 1ap−2

is applied after

odd denominator p − 2 (with p−12 is odd), then

π

4= 1 −

13

+15−

17

. . . −1

p − 2+

1ap−2

. (43)

On the other hand, if the correction term lap

, is applied after the odddenominator p, then

π

4= 1 −

13

+15−

17

. . . −1

p − 2+

1p−

1ap

. (44)

If the correction terms are exact, then both should yield the sameresult. That is,

1ap−2

=1p−

1ap

or1

ap−2+

1ap

=1p

, (45)

is the condition for the end-correction to lead to the exact result.

42


Need for the end-correction terms

◮ The series for π

4 is an extremely slowly convergent series.◮ To obtain value of π which is accurate to 4-5 decimal

places we need to consider millions of terms.◮ To circumvent this problem, Madhavaseems to have found

an ingenious way called “antya-sam. skara”◮ It essentially consists of –

◮ Terminating the series are a particular term if you getboredom (jamitaya).

◮ Make an estimate of the remainder terms in the series◮ Apply it (+vely/-vely) to the value obtained by summation

after termination.

◮ The expression provided to estimate the remainder termsis noted to be quite effective.

◮ Even if a consider a few terms (say 20), we are able to getπ values accurate to 8-9 decimal places.

43

Error-minimization in the evaluation of Pi

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0N11−01

01−01

9−01

8−01

7−01

6−01

5−01

4−01

E rrork 1k 2

44

Construction of the Sine-table◮ A quadrant is divided into 24 equal parts, so that each arc bit

α = 9024 = 3◦45′ = 225′.

◮ A procedure for finding R sin iα, i = 1, 2, . . . 24 is explicitlygiven. PiNi are known.

◮ The R sines of the intermediate angles are determined byinterpolation (I order or II order).

P

P

P

PPP

O

P

222324

23 3 1N N N N222 N 0

1

2

3

45

Recursion relation for the construction of sine-tableAryabhat. ıya’s algorithm for constructing of sine-table

◮ The content of the verse is equivalent to the relation:

R sin(i + 1)α − R sin iα = R sin iα − R sin(i − 1)α −R sin iαR sin α

.

◮ In fact, the values of the 24 Rsines themselves are explicitlynoted in another verse.

◮ The exact recursion relation for the Rsine differences is:

R sin(i+1)α−R sin iα = R sin iα−R sin(i−1)α−R sin iα 2(1−cos α).

◮ Approximation used by Aryabhat.a is 2(1 − cos α) = 1225 .

◮ While, 2(1 − cos α) = 0.0042822, 1225 = 0.00444444.

◮ In the recursion relation provided by Nılakan. t.ha we find1

225 → 1233.5 (= 0.0042827) .

46

Infinite series for the sine function

◮ The verses giving the ∞ series for the sine function is14 –;a.na:h:tya .. a.a:pa:va:geRa:Na .. a.a:pMa ta.�a:tP+l;a:a.na .. a Áh:=e ;t,a .sa:mUa:l+yua:gva:gERaH ;�a�a.$ya.a:va:gRa:h:tEaH kÒ +.ma.a:t,a Á Á.. a.a:pMa :P+l;a:a.na .. a.a:Da.eaY:Da.ea nya:~ya.ea:pa:yRua:pa:�a= tya.jea:t,a Á.j�a.a:va.a:�yEa, .sa:ñÍç ÅÅ*:" +h.eaY:~yEa:va ;�a.va:dõ .a:a.na:tya.a:�a.d:na.a kx +.taH Á Á◮ N0 = Rθ D0 = 1◮ N1 = Rθ × (Rθ)2 Ni+1 = Ni × (Rθ)2

◮ D1 = R2(2 + 22) Di = Di−1 × R2(2i + (2i)2)

◮ .j�a.a:va.a = N0D0

− [N1D1

− (N2D2

− {N3D3

− . . . })]

◮ .j�a.a:va.a:�yEa = For obtaining the jıva (Rsine)

14Yuktidıpika (16th cent) and attributed to Madhava (14th cent. AD).47

Infinite series for the sine function

◮ Expressing the series using modern notation as describedas described in the above verse –

Jıva = Rθ −Rθ × (Rθ)2

R2(2 + 22)+

Rθ × (Rθ)2 × (Rθ)2

R2(2 + 22) R2(4 + 42)− . . .

◮ Simplifying the above we have –

Jıva = Rθ−(Rθ)3

R2 × 6+

(Rθ)5

R4 × 6 × 20−

(Rθ)7

R6 × 6 × 20 × 42+ . . .

◮ Further simplifying –

Jıva = R(

θ −θ3

3!+

θ5

5!−

θ7

7!+ . . .

)

= R sin θ

◮ Thus the given expression ≡ well known sine series.

48

The importance of infinite series in the development ofmodern science

◮ The pivotal role played by Newtonian physics in thedevelopment of modern science is too well known.

◮ That the discovery of infinite series expansion is the key toattain pinnacle to this has been succinctly put forth byV. I. Arnol’d15:

Newton’s basic discovery was that everything had tobe expanded in infinite series . . . Newton, although didnot strictly prove convergence, had not doubts about it. . . What did Newton do in analysis? What was hismain mathematical discovery? Newton invented Taylorseries, the main instrument of analysis.16

15One of the greatest mathematicians of recent times.16V. I. Arnol’d, Barrow and Huygens, Newton and Hooke, trans.

E. J. E. Primrose, Birkhauser Verlag, Basel, 1990, pp.35-42. Cited byC. K. Raju, PHISPC, Cultural Foundations of Mathematics Part 4, PearsonLongman, 2007, p.xxxv.

49

Instantaneous velocity of a planetThe mandaphala or “equation of centre” correction

◮ P0 – mean planet

◮ P – true planet

◮ θ0 – mean longitude

◮ θMS – true longitudecalled themanda-sphut.a.

ϖ

θ θ

Γ

A

O

P

0 MS

0

(direction of mandocca)(planet)

P 0

Q

θ − ϖ

◮ The true longitude of the planet is given by

θ = θ0 ± sin−1( r

Rsin M

)

where M (manda-kendra) = θ0− longitude of apogee

◮ The second term in the RHS, known as manda-phala, takes careof the eccentricity of the planetary orbit.

50

Instantaneous velocity of a planetDerivative of sin−1 function

The instantaneous velocity of the planet called tatkalikagati isgiven by Nılakan. t.ha in his Tantrasangraha as follows:.. a:ndÒ ;ba.a:hu :P+l+va:gRa:Za.ea:a.Da:ta:�a�a.$ya:k+a:kx +.a.ta:pa:de :na .sMa:h:=e ;t,a Áta.�a k+ea:a.f:P+l+a.l+a.�a:k+a:h:ta.Ma :ke +.ndÒ ;Bua:a.�+.�a=;h ya:�a l+Bya:tea Á ÁIf M be the manda-kendra, then the content of the above versecan be expressed as

ddt

[

sin−1( r

Rsin M

)]

=

rR

cos MdMdt

√

1 −( r

R sin M)2

(46)

51

Instantaneous velocity of a planetDerivative of the ratio of two functions

Some of the astronomers in the Indian tradition including Munjala hadproposed the expression for mandaphala to be

∆θ =

rR

sin M(

1 −rR

cos M) , (47)

According to Acyuta, the correction to the mean velocity of a planet toobtain its instantaneous velocity in this case is given by

( rR

cos M)

+

rR

sin M!2

1−rR

cos M!

(

1 −rR

cos M)

dMdt

, (48)

which is nothing but the derivative of (47).

52

Concluding Remarks

◮ It is clear that major discoveries in the foundations of calculus,mathematical analysis, etc., did take place in Kerala School(14-16 century).

◮ Besides arriving at the infinite series, that the Keralaastronomers could manipulate with them to obtain several formsof rapidly convergent series is indeed remarkable.

◮ While the procedure by which they arrived at many of theseresults are evident, there are still certain grey areas (derivativeof sine inverse function, ratio of two functions)

◮ Many of these achievements are attributed to Madhava, wholived in the 14th century (his works ?).

◮ Whether some of these results came to be known to theEuropean mathematicians ? ? . . . .

53

Thanks!

Isn’t enough for one Lecture ?

THANK YOU !

54

Need for the advent of Calculus in India: A Journey into ... · Need for the advent of Calculus in India: A Journey into the 15th Century K. Ramasubramanian Cell for Indian Science

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